MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  infcntss Structured version   Visualization version   GIF version

Theorem infcntss 8092
Description: Every infinite set has a denumerable subset. Similar to Exercise 8 of [TakeutiZaring] p. 91. (However, we need neither AC nor the Axiom of Infinity because of the way we express "infinite" in the antecedent.) (Contributed by NM, 23-Oct-2004.)
Hypothesis
Ref Expression
infcntss.1 𝐴 ∈ V
Assertion
Ref Expression
infcntss (ω ≼ 𝐴 → ∃𝑥(𝑥𝐴𝑥 ≈ ω))
Distinct variable group:   𝑥,𝐴

Proof of Theorem infcntss
StepHypRef Expression
1 infcntss.1 . . 3 𝐴 ∈ V
21domen 7827 . 2 (ω ≼ 𝐴 ↔ ∃𝑥(ω ≈ 𝑥𝑥𝐴))
3 ensym 7864 . . . . 5 (ω ≈ 𝑥𝑥 ≈ ω)
43anim2i 590 . . . 4 ((𝑥𝐴 ∧ ω ≈ 𝑥) → (𝑥𝐴𝑥 ≈ ω))
54ancoms 467 . . 3 ((ω ≈ 𝑥𝑥𝐴) → (𝑥𝐴𝑥 ≈ ω))
65eximi 1750 . 2 (∃𝑥(ω ≈ 𝑥𝑥𝐴) → ∃𝑥(𝑥𝐴𝑥 ≈ ω))
72, 6sylbi 205 1 (ω ≼ 𝐴 → ∃𝑥(𝑥𝐴𝑥 ≈ ω))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  wex 1694  wcel 1975  Vcvv 3168  wss 3535   class class class wbr 4573  ωcom 6930  cen 7811  cdom 7812
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-8 1977  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585  ax-sep 4699  ax-nul 4708  ax-pow 4760  ax-pr 4824  ax-un 6820
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2457  df-mo 2458  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-ral 2896  df-rex 2897  df-rab 2900  df-v 3170  df-dif 3538  df-un 3540  df-in 3542  df-ss 3549  df-nul 3870  df-if 4032  df-pw 4105  df-sn 4121  df-pr 4123  df-op 4127  df-uni 4363  df-br 4574  df-opab 4634  df-id 4939  df-xp 5030  df-rel 5031  df-cnv 5032  df-co 5033  df-dm 5034  df-rn 5035  df-res 5036  df-ima 5037  df-fun 5788  df-fn 5789  df-f 5790  df-f1 5791  df-fo 5792  df-f1o 5793  df-er 7602  df-en 7815  df-dom 7816
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator